BN2 AND SH360 HAD AN NMAC OVER RWY 35 AT LIH.

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I WAS APCHING LIH FROM THE E VFR. I CALLED THE TWR 15 MI OUT. THE SHORTS WAS ON THE ILS TO RWY 35 PLANNING A MISSED APCH TO KREEN AS PUBLISHED. 2 B737'S WERE BEHIND THE SHORTS ON VOR TO RWY 35 APCHS. WHEN THIS CTLR IS FACED WITH TFC FOR RWY 35 FROM THE S OR SE AND VFR TFC FROM THE E; HE REQUIRES THE VFR TFC TO OVERFLY THE FIELD AND ENTER ON L BASE FOR RWY 3. THIS WORKS AS LONG AS THE RWY 35 TFC DOESN'T GO AROUND. IN THIS CASE THE SHORTS WAS PLANNING THE MISS AS PART OF THEIR TRAINING SEQUENCE. THE CTLR DOESN'T ACCEPT OTHER SUGGESTIONS. 3 MI E OF THE FIELD; I HAD THE SHORTS IN SIGHT; ALSO 2-3 MI OUT. I MAINTAINED 1500 FT AGL INSTEAD OF 800 FT PATTERN ALT BECAUSE OF CONCERN FOR HIS RAPID CLB ON THE MISS. I REQUESTED R TFC FOR RWY 3 TO PASS BEHIND THE SHORTS. I WAS AGAIN TOLD TO CROSS THE FIELD FOR L BASE. AS THE SHORTS EXECUTED HIS MISSED APCH; I SAW WE WOULD BE VERY CLOSE. I INITIATED AN IMMEDIATE CLB. THE SHORTS PASSED JUST BENEATH US. IN RETROSPECT; KNOWING THIS CTLR'S RIGIDITY; I SHOULD HAVE TIMED MY PATTERN TO BE WELL BEHIND THE SHORTS. HOWEVER; SINCE I FIRST SAW HIM AT 3 MI AND THE BN2 HAS A SMALL SPD RANGE; THERE WAS LITTLE I COULD HAVE DONE AT THAT POINT. CALLBACK CONVERSATION WITH RPTR REVEALED THE FOLLOWING INFO: PLT WAS ON A FLT FROM OGG TO LIH. HE OPERATES AS A SINGLE PLT PART 135 OP. HE STATES THE TWR IS MANNED BY CONTRACT CTLRS. HE IS CONCERNED WITH A CERTAIN CTLR'S INFLEXIBILITY. WHEN HE HEARS THIS VOICE; HE KNOWS THAT THE ONLY APCH CLRNC HE WILL RECEIVE IS FLY OVERHEAD THE FIELD. IN THIS CASE; SINCE HE WAS COMING FROM THE E; HE SUGGESTED AN ALTERNATE APCH BECAUSE OF CONFLICT WITH A TRAINING ACFT ON THE ILS. WHEN THE CTLR PROVED INFLEXIBLE; THE PLT CONTACTED THE SHORTS ON CTR FREQ TO COORDINATE THEIR APCH. THE SHORTS WOULD CLB SLOWER THAN NORMAL ON HIS MISS; AND THE ISLANDER WOULD SHALLOW HIS DSCNT TO PATTERN ALT. THE SHORTS WAS NOT YET ON TWR FREQ. EVEN WITH THIS COORD; THERE WAS STILL AN NMAC AS THE SHORTS TURNED RIGHT INTO THE ISLANDER WHEN HE MADE HIS TURN TO KREEN.